Communications in Algebra

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Communications in Algebra

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Extension Algebras of Standard Modules

Liping Li ^a

a School of Mathematics , University of Minnesota , Minneapolis , Minnesota , USA Published online: 21 Jun 2013.

To cite this article: Liping Li (2013) Extension Algebras of Standard Modules, Communications in Algebra, 41:9, 3445-3464, DOI: 10.1080/00927872.2012.688155

To link to this article: http://dx.doi.org/10.1080/00927872.2012.688155

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EXTENSION ALGEBRAS OF STANDARD MODULES

Liping Li

School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

LetAbe a basic finite-dimensionalk-algebra standardly stratified for a partial order andbe the direct sum of all standard modules. In this article, we study the extension algebra =Ext^∗_A of standard modules, characterize the stratification property offorandôp, and obtain a sufficient condition forto be a generalized Koszul algebra (in a sense which we define).

Key Words: Extension algebras; Koszul; Quasi-hereditary; Standardly stratiﬁed.

2010 Mathematics Subject Classiﬁcation: 16G10; 16E40.

LetAbe a basic finite-dimensionalk-algebra standardly stratified with respect to a poset indexing all simple modules (up to isomorphism), be the direct sum of all standard modules, andbe the category of finitely generated A-modules with -filtrations. That is, for each M∈, there is a chain 0= M₀⊆M₁⊆ · · · ⊆M_n=M such that M_i/M_i−1 is isomorphic to an indecomposable summand of , 1in. Since standard modules of A are relative simple in , we are motivated to exploit the extension algebra=Ext^∗_A of standard modules. These extension algebras were studied in [1, 6, 11, 19, 23]. In this article, we are interested in the stratification property of with respect toandôp, and its Koszul property since has a natural grading. A particular question is that in which case it is ageneralized Koszul algebra, i.e., ₀ has a linear projective resolution.

By Gabriel’s construction, we associate a locally finite k-linear category to the extension algebra such that the category -mod of finitely generated left -modules is equivalent to the category of finitely generatedk-linear representations of . We show that the category is a directed category with respect to . That is, the morphism spacex y=0 wheneverxy. With this terminology, we have the following theorem.

Theorem 0.1. If A is standardly stratified for , then is a directed category with respect to and is standardly stratified for ôp. Moreover, is standardly stratified for if and only if for all ∈ and s0,Ext^s_A is a projective End_A-module.

Received April 7, 2012; Revised April 13, 2012. Communicated by D. Zacharia.

Address correspondence to Liping Li, School of Mathematics, University of Minnesota, 206 Church St. SE 504 Vincent Hall, Minneapolis, MN 55455, USA; E-mail: [email protected]

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Downloaded by [Bibliothèques de l'Université de Montréal] at 09:07 03 February 2015

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In particular, if A is a quasi-hereditary algebra, then is quasi-hereditary with respect to both and ^op. We also generalize the above theorem to abstract stratifying systemsandExt-projective stratifying systems(EPSS) described in [7, 17, 18, 24].

In the case that the standardly stratiﬁed algebraAis a graded algebra andA₀ is semisimple, its Koszul duality has been studied in [2, 3, 20–22]. Since the extension algebra =Ext^∗_A has a natural grading, and₀=End_A in general is not semisimple, we study the general Koszul property of by using some generalized Koszul theories developed in [10, 12, 14–16], where the degree 0 parts of graded algebras are not required to be semisimple.

Take a fixed EPSS Q. As an analogue to linear modules of graded algebras, we definelinearly filtered modules in this system. With this terminology, a sufficient condition can be obtained for to be a generalized Koszul algebra.

Theorem 0.2. Let Q be an EPSS indexed by a finite poset such that Extⁱ_AQ =0 for all i1 and Hom_AQ Hom_A . Suppose that all are linearly filtered for ∈. If M ∈ is linearly filtered, then the graded -moduleExt^∗_AM has a linear projective resolution. In particular, =Ext^∗_A is a generalized Koszul algebra.

The article is organized as follows. In Section 1 we characterize the stratification property of; In Section 2 we define linearly filtered modules, study their basic properties, and prove the second theorem.

Throughout this article, A is a finite-dimensional basic associative k-algebra with identity 1, where kis algebraically closed. We only consider finitely generated modules and denote by A-mod the category of finitely generated left A-modules.

Maps and morphisms are composed from right to left.

1. STRATIFICATION PROPERTY OF EXTENSION ALGEBRAS

Let be a ﬁnite preordered set parameterizing all simple A-modules S (up to isomorphism). This preordered set also parameterizes all indecomposable projective A-modules P (up to isomorphism). According to [4], the algebra A is standardly-stratiﬁed with respect to if there exist modules , ∈, such that the following conditions hold:

(1) The composition factor multiplicity S=0 whenever; and

(2) For every∈, there is a short exact sequence 0→K→P→→0 such thatKhas a ﬁltration with factors , where > .

In some literatures, the preordered setis supposed to be a poset ([5]) or even a linearly ordered set ([2, 3]). Algebras standardly stratiﬁed in this sense are called strongly standardly stratiﬁed([8, 9]). In this articleis supposed to be a poset.

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Actually, thosestandard modulescan be deﬁned as¹ =P/

tr_PP

where tr_PP is the trace of P in P. See [5, 24] for more details. Let be the direct sum of all standard modules andbe the full subcategory ofA-mod such that each object inhas a filtration by standard modules. ThenAis standardly stratified for if and only if _AA∈, or equivalently, every indecomposable projectiveA-module has a filtration by standard modules.

Throughout this section, we suppose thatAis standardly stratiﬁed with respect to if it is not speciﬁed. We also remind the reader that is closed under extensions, kernels of epimorphisms, and direct summands, but is not closed under cokernels of monomorphisms.

Given M ∈ and a fixed filtration 0=M₀⊆M₁⊆ · · · ⊆M_n =M, we define thefiltration multiplicitym=M to be the number of factors isomorphic to in this filtration. By Lemma 1.4 of [7], the filtration multiplicities defined above are independent of the choice of a particular filtration. Moreover, since each standard module has finite projective dimension, we deduce that every A-module contained inhas finite projective dimension. Therefore, the extension algebra =Ext^∗_A is finite-dimensional.

Lemma 1.1. Let, be standard modules. ThenExtⁿ_A =0iffor all n0.

Proof. First, we claimⁱ =0 whenever for all i0, where is the Heller operator. Indeed, fori=0 the conclusion holds clearly. Suppose that it is true for allin, and considerⁿ⁺¹. We have the following exact sequence:

0−→ⁿ⁺¹−→P−→ⁿ−→0

By the induction hypothesis, ⁿ =0 whenever . Therefore, P =0 whenever , and hence ⁿ⁺¹ =0 whenever . The claim is proved by induction.

The above short exact sequence induces a surjection Hom_Aⁿ → Extⁿ_A . Thus it sufﬁces to show Hom_Aⁿ =0 for alln0 if. By the above claim, all ﬁltration factorsofⁿsatisfy, and hence. But Hom_A =0 whenever. The conclusion follows.

Gabriel’s construction gives rise to a bijective correspondence between finite- dimensional algebras and locally finite k-linear categories with finitely many objects. Explicitly, to each finite-dimensional k-algebra A with a chosen set of orthogonal primitive idempotents e_∈ satisfying

∈e=1, we define a k- linear category with Ob =e_∈ and e e=eAeHom_AAe Ae. Conversely, given a locally finite k-linear category with finitely many objects,

1In [2, 3] standard modules are deﬁned as=P/

>tr_P

P. Note that in their setup is a linear order, so this description of standard modules coincides with ours.

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we deﬁne A=

xy∈Obx y, and the multiplication in A is induced by the composition of morphisms in in an obvious way. Clearly, A-mod is Morita equivalent to the category of all ﬁnite-dimensional k-linear representations of . We then calltheassociated categoryofA andAtheassociated algebraof . The categoryis calleddirectedif there is a partial orderon Obsuch thatx y= 0 unlessxy. See [12, 13] for more details.

Now let=Ext^∗_A . This is a graded ﬁnite-dimensional algebra equipped with a natural grading. In particular, ₀=End_A. For each ∈, is an indecomposable A-module. Therefore, up to isomorphism, the indecomposable projective -modules are exactly those Ext^∗_A ,∈.

The associatedk-linear category of has the following structure: Ob= _∈; the morphism space =Ext^∗_A . The partial orderinduces a partial order on Ob, which we still denote by, namely, if and only if .

Proposition 1.2. The associated category of is directed with respect to . In particular, is standardly stratiﬁed with respect to^op and all standard modules are projective.

Proof. The ﬁrst statement follows from the previous lemma. The second statement is also clear. Indeed, since is directed with respect to,eeHomQ Q= 0 if, whereQ Qare projective-modules. Thus tr_QQ=0 whenever , or equivalently, tr_QQ=0 whenever^op. Therefore, all standard modules

with respect to ^opare projective.

The following proposition characterizes the stratiﬁcation property of ak-linear category directed with respect to.

Proposition 1.3. Let be a locally ﬁnite k-linear category directed with respect to a partial order on Ob . Then it is stratiﬁed for this order. The standard modules are isomorphic to indecomposable summands of

x∈Obx x. Moreover, this stratiﬁcation is standard if and only if for each pair of objectsx y∈Ob,x y is a projectivey y-module.

Proof. This is just a collection of results in [12]. The ﬁrst statement is Corollary 5.4;

the second statement comes from Proposition 5.5; and the last statement is

Theorem 5.7.

Now we restate and prove the ﬁrst theorem.

Theorem 1.4. IfA is standardly stratified for , then is a directed category with respect to and is standardly stratified for ôp. Moreover, is standardly stratified for if and only if for all ∈ and s0, Ext^s_A is a projective End_A-module.

Proof. The ﬁrst statement follows from Proposition 1.2, and the second statement

follows from Proposition 1.3.

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In the case thatAis quasi-hereditary, we have the following corollary.

Corollary 1.5. If Ais a quasi-hereditary algebra with respect to, then is quasi- hereditary with respect to bothand^op.

Proof. We have showed that is standardly stratiﬁed with respect to^opand the corresponding standard modules1for∈. Therefore,

End=End111=Ext^∗_A =End_Ak sinceAis quasi-hereditary. So is also quasi-hereditary with respect to^op.

Now consider the stratiﬁcation property ofwith respect to. The associated category is directed with respect to . Since Ext^∗_A =End_Ak for all ∈, =Ext^∗_A is a projective k-module for each pair ∈ . Therefore, is standardly stratiﬁed forby the previous theorem. Moreover, by Proposition 1.3, the standard modules of (or the standard modules of ) are precisely indecomposable summands of

∈Ext^∗_A

∈k. Clearly, for ∈, Endk kk, so is quasi-hereditary with respect to. The following example from 8.2 in [9] illustrates why we should assume that is a partial order rather than a preorder. Indeed, in a preordered set, we cannot deducex=y ifxy andyx.

Example 1.6. Let A be the path algebra of the following quiver with relations ₁₁=₂₂=₂₁=₁₂=0. Deﬁne a preorder by letting xy < z and y x < z:

Projective modules and standard modules are described as follows:

P_x_x= x y x

P_y= y x z

y

_y=y

x P_z_z=z y

Then the associated category of =Ext^∗_A is not a directed category since both Hom_A_x _yand Hom_A_y _xare nonzero.

Now we generalize the above results to EPSS. From now on the algebraA is finite-dimensional and basic, but we do not assume that it is standardly stratified for some partial order, as we did before. The EPSS we describe in this article is indexed by a finite posetrather than a linearly ordered set as in [17, 18]. However, this difference is not essential, and all properties described in [17, 18] can be applied to our situation with suitable modifications.

Deﬁnition 1.7(Deﬁnition 2.1 in [18]). Let = _∈ be a set of nonzero A-modules andQ=Q_∈ be a set of indecomposableA-modules, both of which

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are indexed by a ﬁnite poset . We call Q an EPSS if the following conditions are satisﬁed:

(1) Hom_A =0 if;

(2) For each ∈, there is an exact sequence 0→K→Q→ →0 such that K has a ﬁltration only with factors isomorphic to satisfying > ;

(3) For everyA-moduleM∈ and∈, Ext¹_AQ M=0.

We denote andQ the direct sums of all ’s andQ’s, respectively,∈. Given an EPSS Q indexed by , is a stratifying system (SS): Hom_A =0 if, and Ext¹_A =0 if≮. Conversely, given a stratifying system , we can construct an EPSS Q unique up to isomorphism. See [18] for more details. Moreover, as described in [18], the algebra B=End_AQ^opis standardly stratiﬁed, and the functor e_Q=Hom_AQ−gives an equivalence of exact categories between and_B.

To study the extension algebra =Ext^∗_A , one may want to use projective resolutions of . However, different from the situation of standardly stratiﬁed algebras, the regular module_AAin general might not be contained in . If we suppose that_AAis contained in (in this case, the stratifying system is said to bestandard) and is closed under the kernels of surjections, then by Theorem 2.6 in [17] A is standardly stratiﬁed for , and those ’s coincide with standard modules of A. This situation has been completely discussed previously.

Alternately, we use the relative projective resolutionswhose existence is guaranteed by the following proposition.

Proposition 1.8(Corollary 2.11 in [18]). Let Qbe an EPSS indexed by a ﬁnite poset. Then for eachM∈ , there is a ﬁnite resolution

0−→Q^d−→ · · · −→Q⁰−→M−→0

such that each kernel is contained in , where0=Qⁱ∈addQfor0id.

The numberdin this resolution is called therelative projective dimensionofM.

Proposition 1.9. Let Q be an EPSS indexed by a ﬁnite poset and dbe the relative projective dimension of . IfExt^s_AQ =0for alls1, then forM N ∈ ands > d,Ext^s_AM N=0.

Proof. Since both M and N are contained in , it is enough to show that Ext^s_A =0 for all s > d. If d=0, then Q= , and the conclusion holds trivially. So we suppose d1. Applying the functor Hom_A− to the exact sequence

0−→K₁−→Q−→ −→0 we get a long exact sequence. In particular, from the segment

Ext^s−1_A Q −→Ext^s−1_A K₁ −→Ext^s_A −→Ext^s_AQ

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of this long exact sequence we deduce that Ext^s_A Ext^s_A⁻¹K₁ since the ﬁrst and last terms are 0. Now applying Hom_A− to the exact sequence

0−→K₂−→Q¹−→K₁−→0

we get Ext^s−_A¹K₁ Ext^s−_A²K₂ . Thus Ext^s_A Ext^s−d_A K_d by induction. ButK_dQ^d∈addQ. The conclusion follows.

Thus =Ext^∗_A is a ﬁnite-dimensional algebra under the given assumption.

There is a natural partition on the ﬁnite poset as follows. Let ₁ be the subset of all minimal elements in,₂be the subset of all minimal elements in \₁, and so on. Then= i1_i. With this partition, we can introduce aheight functionh →in the following way: For∈_i⊆,i1, we deﬁneh=i.

For each M∈ , we define suppM to be the set of elements ∈ such that M has a -filtration in which there is a factor isomorphic to . For example, supp =. By Lemma 2.6 in [18], the multiplicities of factors ofM is independent of the choice of a particular -filtration. Therefore, suppMis well defined. We also define minM=minh∈suppM.

Lemma 1.10. Let Q be an EPSS indexed by a ﬁnite poset . For each M∈ , there is an exact sequence 0→K₁→Q⁰→M such that K₁∈ and minK₁ >minM, whereQ⁰∈addQ.

Proof. This is Proposition 2.10 in [18] which deals with the special case thatis a linearly ordered set. The general case can be proved similarly by observing the fact

that Ext¹_A =0 if h=h.

By this lemma, the relative projective dimension of every M∈ cannot exceed the length of the longest chain in.

As before, we let be thek-linear category associated to=Ext^∗_A . Theorem 1.11. Let Q be an EPSS indexed by a finite poset . Such that Extⁱ_AQ =0 for all i1. Then is a directed category with respect to and is standardly stratified forôp. Moreover, it is standardly stratified forif and only if for alls0,Ext^s_A is a projectiveEnd_A -module, ∈.

Proof. We only need to show thatis a directed category with respect tosince the other statements can be proved as in Theorem 1.4. We know Hom_A =0 if and Ext¹_A =0 for all≮. Therefore, it sufﬁces to show that for alls2, Ext^s_A =0 if ≮.

By Proposition 1.8 and Lemma 1.10, has a relative projective resolution 0−→Q^d−→ · · ·^f^d −→^f¹ Q⁰−→^f⁰ −→0

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such that for each mapf_t, min(K_t >minK_t−1, whereK_t =Kerf_tand 1td.

By Proposition 1.9, Ext^s_A =0 ifs > d; if 2sd, we have Ext^s_A Ext¹_AK_s₋₁ . But we have chosen

minK_s−1 >minK_s−2 >· · ·>min =hh

Thus each factor appearing in a -ﬁltration of K_s₋₁ satisﬁesh > h, and hence. Since Ext¹_A =0 for all, we deduce

Ext^s_A Ext¹_AK_s−₁ =0

This ﬁnishes the proof.

The following corollary is a generalization of Corollary 1.5.

Corollary 1.12. Let Q be an EPSS indexed by a ﬁnite poset . If for all s1 and ∈, we have Ext^s_AQ =0 and End_A k, then is quasi- hereditary with respect to bothand^op.

Proof. This can be proved as Corollary 1.5.

2. KOSZUL PROPERTY OF EXTENSION ALGEBRAS

There is a well-known duality related to the extension algebras: the Koszul duality. Explicitly, if A is a graded Koszul algebra with A₀ being a semisimple algebra, then B=Ext^∗_AA₀ A₀ is a Koszul algebra, too. Moreover, the functor Ext^∗_A− A₀gives an equivalence between the category of linearA-modules and the category of linear B-modules.² However, even if A is quasi-hereditary with respect to a partial order,Bmight not be quasi-hereditary with respect toor^op. This problem has been considered in [2, 20].

On the other hand, ifAis quasi-hereditary with respect to, we have showed that the extension algebra =Ext^∗_A is quasi-hereditary with respect to both and ôp. But is in general not a Koszul algebra in a sense which we define later. In this section, we want to get a sufficient condition for to be a generalized Koszul algebra.

We work in the context of the EPSS described in last section. Let Qbe an EPSS indexed by a ﬁnite poset;Q=

∈Qand =

∈ .We insist on the following conditions: Ext^s_AQ=0 for all s1; each has a simple top S; and SS for=. These conditions are always true for the classical stratifying system of a standardly stratiﬁed basic algebra. In particular, in that caseQ=AA.

Proposition 2.1. Let 0=M∈ and i=minM. Then there is an exact sequence

0−→M1−→M−→

h=i

⊕m

−→0 (2.1)

such thatM1∈ andminM1 >minM.

2In [12] we generalized these results to the situation thatA₀ is a self-injective algebra.

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Proof. This is Proposition 2.9 in [18], which deals with the special case thatis a linearly ordered set. The general case can be proved similarly by observing the fact

that Ext¹_A =0 if h=h.

It is clear thatm=M . Based on this proposition, we give the following deﬁnition.

Deﬁnition 2.2. LetM ∈ with minM=i. We say M is generated in height iif in Eq. (2.1), we have

TopM=M/radM Top

h=i

⊕m

=

h=i

S^⊕^m

We introduce some notation: IfM∈ is generated in heighti, then deﬁne M_i=

h=i ⊕m

in Eq. (2.1). IfM1is generated in some heightj, we can deﬁne M2=M11andM1_j in a similar way. This procedure can be repeated.

Proposition 2.3. Let0→L→M →N →0be an exact sequence in . IfM is generated in heighti, so is N. Conversely, if both Land N are generated in height i, thenM is generated in height ias well.

Proof. We always have Top N ⊆Top M and Top M⊆Top L⊕Top N. The conclusion follows from these inclusions and the rightmost identity in the above

deﬁnition.

Notice thatQ =1 and Q =0 for all. We claim that Q is generated in height h for ∈. Indeed, the algebra B=End_AQôp is a standardly stratified algebra, with projective modules Hom_AQ Q and standard modules Hom_AQ , ∈. Moreover, the functor Hom_AQ− gives an equivalence between ⊆A-mod and _B⊆B-mod. Using this equivalence and the standard filtration structure of projective B-modules, we deduce the conclusion.

Lemma 2.4. IfM ∈ is generated in heightiwithM =m, thenM has a relative projective coverQⁱ

h=iQ^⊕^m. Proof. There is a surjectionf M→

h=i

⊕m

by Proposition 2.1. Consider the following diagram:

Since Qⁱ is projective in , the projection p factors through the surjection f. In particular, Top _h₌_i ^⊕^m

=

h=iS^⊕^m is in the image of fq. Since M is generated in height i, f induces an isomorphism between Top M and

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Top _h=i ^⊕m

. Thus TopM is in the image of q, and henceq is surjective. It is clear that qis minimal, soQ¹=

h=iQ^⊕m is a relative projective cover ofM.

The uniqueness follows from Proposition 8.3 in [24].

We useⁱMto denote theith relative syzygyofM. Actually, for everyM ∈ , there is always a relative projective cover by Proposition 8.3 in [24].

The following deﬁnition is an analogue oflinear modulesin the representation theory of graded algebras.

Deﬁnition 2.5. An A-module M∈ is said to be linearly ﬁltered if there is somei∈such that ^sMis generated in heighti+sfors0.

Equivalently, M∈ is linearly ﬁltered if and only if it is generated in heightiand has a relative projective resolution

0−→Q^l−→Q^l⁻¹· · · −→Qⁱ⁺¹−→Qⁱ−→M −→0 such that eachQ^sis generated in heights,isl.

We remind the reader that there is a common upper bound for the relative projective dimensions of modules contained in , which is the length of the longest chains in the finite poset . It is also clear that if M is linearly filtered, so are all relative syzygies and direct summands. In other words, the subcategory constituted of linearly filtered modules contains all relative projective modules, and is closed under summands and relative syzygies. But in general it is not closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms.

Proposition 2.6. Let 0→L→M →N →0 be an exact sequence in such that all terms are generated in heighti. IfLis linearly filtered, thenMis linearly filtered if and only ifN is linearly filtered.

Proof. Letm=M ,l=L , andn=N . By the previous lemma, we get the following commutative diagram:

Since Lis generated in heighti+1, by Proposition 2.3, Mis generated in heighti+1 if and only if Nis generated in heighti+1. ReplacingL,M, andN

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by L, M, and N, respectively, we conclude that²Mis generated in heighti+2 if and only if²Nis generated in heighti+2. The conclusion follows

from induction.

Corollary 2.7. Suppose thatM ∈ is generated in height i and linearly ﬁltered.

If

h=i is linearly ﬁltered, then M1 is generated in height i+1 and linearly ﬁltered.

Proof. Clearly,

h=i is generated in height i. Let m=M . Notice that bothM and

h=i ⊕m

have projective cover

h=iQ^⊕m . Thus the exact sequence

0−→M1−→M −→

h=i

⊕m −→0 induces the following diagram:

Consider the top sequence. Since both

h=i ⊕m

and M are generated in heighti+1 and linearly ﬁltered,M1is also generated in heighti+1

and linearly ﬁltered by Propositions 2.3 and 2.6.

These results tell us that linearly ﬁltered modules have properties similar to those of linear modules of graded algebras.

Lemma 2.8. Let M ∈ be generated in height i and m=M . If Hom_AQ Hom_A , then

Hom_AM Hom_A

h=i

Q^⊕m

Hom_A

h=i

⊕m

Proof. We claim that Hom_AQ Hom_A as End_A -modules implies Hom_AQ Hom_A for every ∈. Then the second isomorphism follows immediately. First, notice that Hom_A is a basic algebra with n nonzero indecomposable summands Hom_A , ∈, where n is the cardinal number of . But Hom_AQ

∈Hom_AQ has at least n nonzero indecomposable summands. If Hom_AQ Hom_A , by the Krull–Schmidt theorem, Hom_AQ must be indecomposable and is isomorphic to some Hom_A .

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If∈ is maximal, Hom_A Hom_AQ sinceQ . Deﬁne₁ to be the subset of maximal elements in , and consider ₁∈\₁ which is maximal. We have

Hom_AQ₁ Hom_A Hom_AQ for every ∈₁ since End_A is a basic algebra. Therefore, Hom_AQ

1 must

be isomorphic to some Hom_A with∈\₁. But Hom_A contains a surjection from to the direct summand of , and Hom_AQ₁ contains a surjection fromQ

1 to if and only if₁=. Thus we get₁=. Repeating the above process, we have Hom_AQ Hom_A for every∈.

Applying Hom_A− to the surjectionM →

h=i

⊕m

, we get Hom_A

h=i

⊕m

⊆Hom_AM

Similarly, from the relative projective covering map

h=iQ^⊕m →M, we have

Hom_AM ⊆Hom_A

h=i

Q^⊕m

Comparing these two inclusions and using the second isomorphism, we deduce the

ﬁrst isomorphism.

The reader may be aware that the above lemma is an analogue to the following result in representation theory of graded algebras: If A is a graded algebra andM is a graded module generated in degree 0, then Hom_AM A₀Hom_AM₀ A₀. Lemma 2.9. Suppose that Hom_AQ Hom_A . IfM∈ is generated in heighti, thenExt^s_AM Ext^s−_A¹ M for alls1.

Proof. Letm=M . Applying Hom_A− to the exact sequence

0−→ M−→

h=i

Q^⊕^m −→M −→0

we get a long exact sequence. In particular, for alls2, by observing the segment 0 = Ext^s−1_A

h=i

Q^⊕m

→Ext^s−1_A M

→Ext^s_AM →Ext^s_A

h=i

Q^⊕^m

=0

we conclude Ext^s−_A¹ M Ext^s_AM .

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Fors=1, we have

0→Hom_AM →Hom_A

h=i

Q^⊕^m

→Hom_A M →Ext¹_AM →0

By the previous lemma, the ﬁrst inclusion is an ismorphism. Thus Ext¹_AM

Hom_A M .

We remind the reader that, although =Ext^∗_A has a natural grading, the classical Koszul theory cannot be applied directly since₀=End_A may not be a semisimple algebra. Thus we introducegeneralized Koszul algebrasas follows.

Deﬁnition 2.10. Let R=

i0R_i be a positively graded locally ﬁnite k-algebra, i.e., dim_kR_i< for eachi0. A gradedR-moduleM is said to be linear if it has a linear projective resolution

· · · −→P^s−→ · · · −→P¹−→P⁰−→M

such thatP^sis generated in degrees. The algebraRis said to be generalized Koszul ifR₀viewed as aR-module has a linear projective resolution.

It is easy to see from this deﬁnition thatM is linear if and only if ^sM is generated in degreesfor alls0.

Proposition 2.11. Suppose that Hom_AQ Hom_A , and are linearly ﬁltered for all∈. IfM ∈ is linearly ﬁltered, then

Extⁱ_A⁺¹M =Ext¹_A ·Extⁱ_AM

for all i0, i.e., Ext^∗_AM as a graded =Ext^∗_A -module is generated in degree 0.

Proof. Suppose that M is generated in height d and linearly ﬁltered. By Lemma 2.9,

Extⁱ_A⁺¹M Extⁱ_A M

But is generated in heightd+1 and linearly ﬁltered. Thus by induction Extⁱ⁺¹_A M Ext¹_AⁱM Extⁱ_AM Hom_AⁱM Therefore, it sufﬁces to show

Ext¹_AM =Ext¹_A ·Hom_AM since we can replaceM byⁱ M, which is linearly ﬁltered as well.

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Letm=M and deﬁneQ⁰=

h=dQ^⊕^m,M₀=

h=d

⊕m

. We have the following commutative diagram:

whereQ⁰1= M₀, see Proposition 2.1.

Observe that all terms in the top sequence are generated in heightd+1 and linearly ﬁltered. For every∈withh=d+1, we have

M +M1 =Q⁰1

Let r s andt be the corresponding numbers in the last equality. Then we get a split short exact sequence

0−→

h=d+1

⊕r

−→

h=d+1

⊕t

−→

h=d+1

⊕s −→0

Applying Hom_A− to this sequence and using Lemma 2.8, we obtain the exact sequence

0→Hom_AM1 →Hom_AQ⁰1 →Hom_A M →0 Therefore, each map M→ can extend to a mapQ⁰1→ .

To prove Ext¹_AM =Ext¹_A ·Hom_AM , by Lemma 2.9 we ﬁrst identify Ext¹_AM with Hom_A M . Take an element x∈Ext¹_AM and letg M→ be the corresponding homomorphism. As we just showed,gcan extend toQ⁰1, and hence there is a homomorphismg Q˜ ⁰1→ such that g=

˜

g, whereis the inclusion:

We have the following commutative diagram:

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wherepis the projection ofM onto M₀. The map g Q˜ ⁰1→ gives a push-out of the bottom sequence

SinceM₀

h=d

⊕m

, the bottom sequence represents some y∈Ext¹_A ^⊕m =^m

i=1

Ext¹_A where m=

h=dm. Therefore, we can write y=y₁+ · · · +y_m, where y_i∈ Ext¹_A is represented by the sequence

0−→ −→E_i−→ −→0

Composed with the inclusions → , we get the mapp₁ p_mwhere each componentp_iis deﬁned in an obvious way. Consider the pullbacks:

Denote byx_ithe top sequence. Then x=^m

i=1

x_i=^m

i=1

y_i·p_i∈Ext¹_A ·Hom_AM

so Ext¹_AM ⊆Ext¹_A ·Hom_AM . The other inclusion is obvious.

Now we can prove the main result.

Theorem 2.12. Let Q be an EPSS indexed by a finite poset such that Extⁱ_AQ =0 for alli1 and Hom_AQ Hom_A . Suppose that all are linearly filtered for∈. IfM ∈ is linearly filtered, then the graded module Ext^∗_AM has a linear projective resolution. In particular, =Ext^∗_A is a generalized Koszul algebra.

Proof. Suppose thatM is generated in height d. Deﬁne m=M for∈, Q⁰=

h=dQ^⊕m , andM₀=

h=d ⊕m

. As in the proof of the previous lemma,

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we have the following short exact sequence of linearly ﬁltered modules generated in heightd+1

0−→ M−→ M₀−→M1−→0

where M₀=Q⁰1. This sequence induces exact sequences recursively (see the proof of Proposition 2.6)

0−→ⁱM−→ⁱM₀−→ⁱ⁻¹M1−→0

where all modules are linearly ﬁltered and generated in heightd+i. Again as in the proof of the previous lemma, we get an exact sequence

0→Hom_Aⁱ⁻¹M1 →Hom_Aⁱ M₀ →Hom_AⁱM →0 According to Lemma 2.9, the above sequence is isomorphic to

0→Extⁱ⁻¹_A M1 →Extⁱ_AM₀ →Extⁱ_AM →0 Now let the indexi vary and put these sequences together. We have

0−→EM11 −→EM₀−→^p EM−→0

whereE=Ext^∗_A− and− is the degree shift functor of graded modules. That is, for a graded moduleT =

i0T_i,T1i is deﬁned to beT_i₋₁.

SinceM₀∈add , EM₀is a projective-module. It is generated in degree 0 by the previous lemma. Similarly,EM1is generated in degree 0, soEM11 is generated in degree 1. Therefore, the mappis a graded projective covering map.

Consequently,EMEM11 is generated in degree 1.

ReplacingM byM1(since it is also linearly ﬁltered), we have ²EMEM11EM11 EM22

which is generated in degree 2. By recursion,ⁱEMEMiiis generated in degreei for alli0. Thus EMis a linear-module.

In particular, letM=Q for a certain∈. We get that EQ=Ext^∗_AQ =Hom_AQ is a linear-module. Therefore,

∈

EQ =

∈

Hom_AQ Hom_A

∈

Q

=Hom_AQ Hom_A =₀

is a linear-module. So is a generalized Koszul algebra.

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Remark 2.13. To get the above result we made some assumptions on the EPSS Q. Firstly, each has a simple top S and SS for =; secondly, Ext^s_AQ =0 for every s1. These two conditions always hold for standardly stratiﬁed basic algebras. We also suppose that Hom_A Hom_AQ . This may not be true even ifAis a quasi-hereditary algebra.

Although is proved to be a generalized Koszul algebra, in general it does not have the Koszul duality. Consider the following example.

Example 2.14. Let A be the path algebra of the following quiver with relation ·=0. Put an orderx < y < z.

The projective modules and standard modules ofA are described as follows:

P_x = x y x z

P_y= y

x z P_z=z

_x =x _y=y

x _z=zP_z

This algebra is quasi-hereditary. Moreover, Hom_A Hom_AA , and all standard modules are linearly ﬁltered. Therefore, =Ext^∗_A is a generalized Koszul algebra by the previous theorem.

We explicitly compute the extension algebra . It is the path algebra of the following quiver with relation·=0:

P_x= x₀ y₀ y₁ z₁

P_y=y₀

z₁ P_z=z₀

and

_x=x₀ _y=y₀ _z=z₀ ₀=x₀

y₀⊕y₀⊕z₀

Here we use indices to mark the degrees of simple composition factors. As asserted by the theorem,₀has a linear projective resolution. Butis not a linear-module (we remind the reader that the two simple modulesyappearing inP_xlie in different degrees!).

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By computation, we get the extension algebra =Ext^∗₀ ₀, which is the path algebra of the following quiver with relation·=0:

x−→ y−→ z

Sinceis a Koszul algebra in the classical sense, the Koszul duality holds in. It is obvious that the Koszul dual algebra of is not isomorphic to. Therefore, as we claimed, the Koszul duality does not hold in .

Let us return to the question of whether =Ext^∗_A is standardly stratiﬁed with respect to . According to Proposition 1.3, this happens if and only if for each pair and s0, Ext^s_A is a projective End_A -module.

Putting direct summands together, we conclude that is standardly stratiﬁed with respect to if and only if Ext^s_A is a projective

∈End_A -module.

With the conditions in Theorem 2.12, Ext^s_A Hom_A^s

for alls 0 and ∈ by Lemma 2.9. Notice that ^s is linearly ﬁltered. Suppose that min^s =dandm=^s . Then

Ext^s_A Hom_A^s Hom_A

h=d

⊕m

(2.2)

which is a projective₀=End_A -module.

With this observation, we have the following corollary.

Corollary 2.15. Let Q be an EPSS indexed by a finite poset . Suppose that all are linearly filtered for ∈, and Hom_AQ Hom_A . Then =Ext^∗_A is standardly stratified for if and only if End_A is a projective

∈End_A -module.

Proof. If is standardly stratiﬁed for , then in particular ₀=End_A is a projective

∈End_A -module by Proposition 1.3. Conversely, if ₀= End_A is a projective

∈End_A -module, then by the isomorphism in (2.2) Ext^s_A =

∈Ext^s_A is a projective ₀-module for all s0, so it is a projective

∈End_A -module as well. Again by Proposition 1.3, is standardly

stratiﬁed with respect to .

If A is quasi-hereditary with respect to such that all standard module are linearly ﬁltered, then =Ext^∗_A is again quasi-hereditary for this partial order by Corollary 1.5, and₀ has a linear projective resolution by the previous theorem.

Letbe the direct sum of all standard modules of with respect to. The reader may wonder whether has a linear projective resolution as well. The following proposition gives a partial answer to this question.

Proposition 2.16. With the above notation, if has a linear projective resolution, then₀ , or equivalentlyHom_A =0only if=, ∈. If furthermore Hom_AA End_A, thenA/radA.

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Proof. We have proved that the k-linear category associated to is directed with respect to . By Proposition 1.3, standard modules of for are exactly indecomposable summands of

∈End_A, i.e.,

∈End_A

∈k. Clearly,⊆₀=End_A. If has a linear projective resolution, then by Corollary 2.4 and Remark 2.7 in [12],is a projective₀-module. Consequently, every summand k is a projective ₀-module. Since both and ₀ have exactly pairwise non-isomorphic indecomposable summands, we deduce ₀

∈k, or equivalently Hom_A =0 if =. If furthermore Hom_AA End_A, then

Hom_AA End_A

∈

kA/radA

ACKNOWLEDGMENTS

The author would like to thank his thesis advisor, Professor Peter Webb, for proposing to study the the extension algebras of standard modules, and carefully checking the manuscript. He also thanks Prof. Mazorchuk for pointing out many related works, which are unknown to the author before.

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